Arduino - WiFi connection through serial

The problem is that I cannot find out, how to interface it with Arduino.../

That is why people buy Moteinos. A Moteino board has a 328p with a RFM69/RFM95/RFM96 RF module of your choice soldered on the back.

The boards you link to are probably not going work together. The Nano uses 5V logic levels and the RFM69 board probably uses 3.3V logic levels. I say probably because it looks to be a clone of a HopeRF board but one can never be really sure with clone boards.